640 ORDINARY DIFFERENTIAL EQUATIONS: SCALAR EQUATIONS [24] To find (10) as the solution, write the differential equation as y - a(t)y = b(t) and multiply every term by exp(- J"' a(s) 05): Since the left-hand side of (11) is precisely the derivative of the expression y(t) exp (- J"' a(s) ds), (11) can be rewritten as Integrate both sides of (12) and multiply through by el'" to obtain (10). The expression exp(- J"' a(s) ds) that made the left side of (11) the exact derivative of a function is called an integrating factor. The above four classks of differential equations are called linear differential equations. The first two were autonomous; the second two were not. The first and third -without the b-term - are called homogeneous; the second and fourth, with the b-term, are called nonhomogeneous.

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Unformatted text preview: Example 24.6 (Derivation of Density Functions from Failure Rates) TRtfbe the density function for a continuous random variable t 2 0 and let F be the correspondingstribution function. Think of the random variable t axenoting the lifetime of a mechanical or electrical component. Then, s&) = R(t) = 1 - F(t) = Pr{T > t}, the probability that the component lasts at least t tiine units, is called the reliability function. Given f, F, and R, the failure rate or hazard function Z is defined as The function Z can be thought of as the probability that the.component will fail in 'the next At time units, given that it has not failed up to time t, because the latter conditional probability is equal to Pr(t < T 5 t + At) P r ( t < T < t + A t I T > t ) = P(T > t ) = Z(t) At....
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